Don Reble - cp's OEIS frontend

A368374 a(n) = smallest k such that AM(k) - GM(k) >= n, where AM(k) and GM(k) are the arithmetic and geometric means of [1,...,k].

1, 11, 19, 27, 35, 43, 50, 58, 66, 74, 81, 89, 97, 104, 112, 120, 127, 135, 143, 150, 158, 165, 173, 181, 188, 196, 204, 211, 219, 226, 234, 242, 249, 257, 264, 272, 280, 287, 295, 302, 310, 318, 325, 333, 340, 348, 356, 363, 371, 378, 386, 394, 401, 409, 416

Offset: 0

N. J. A. Sloane, Jan 27 2024, following a suggestion from Don Reble

nonn

The difference d(x) = AM(1,2,3,...,x) - GM(1,2,3,...,x) increases. The first difference of d(x) approaches a limit, 1/2 - 1/e (0.13212...). So we could define a(n) to be the least x such that d(x) >= n. - Don Reble, Jan 27 2024. Which is what I did.

			The values of AM(i)-GM(i) for i = 1, ..., 11 are 0, 0.0857864376269049512, 0.1828794071678603411, 0.2866361605993568152, 0.3948289153026481077, 0.5062048344760910451, 0.6199848408587035501, 0.7356494004968713999, 0.8528337256030871195, 0.9712713118832352378, 1.0907612204156046410, so a(1) = 11.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A068996, A368366.

Digits:=20;
AM := proc(n) local i; add(i,i=1..n)/n; end;
GM := proc(n) local i; mul(i,i=1..n)^(1/n); end;
don := proc(n) evalf(AM(n) - GM(n)); end;
a:=[1]; w:=1;
for i from 1 to 300 do
   if don(i) >= w then a:=[op(a),i]; w:=w+1; fi;
od:
a;

from math import factorial
def A368374(n):
    if n == 0: return 1
    m = (n<<1)-1
    kmin, kmax = m, m
    while factorial(kmax)< (kmax-m)**kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if factorial(kmid)<Chai Wah Wu, Jan 27 2024

a(39)-a(54) from Alois P. Heinz, Jan 27 2024

A355930 Sum of the prime indices of n minus the sum of the prime indices of the smallest number with same prime signature as n, when the sum is taken with multiplicity, as in A056239.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 2, 1, 4, 0, 5, 2, 2, 0, 6, 1, 7, 1, 3, 3, 8, 0, 4, 4, 3, 2, 9, 0, 10, 0, 4, 5, 4, 0, 11, 6, 5, 1, 12, 1, 13, 3, 3, 7, 14, 0, 6, 3, 6, 4, 15, 2, 5, 2, 7, 8, 16, 0, 17, 9, 4, 0, 6, 2, 18, 5, 8, 2, 19, 0, 20, 10, 4, 6, 6, 3, 21, 1, 4, 11, 22, 1, 7, 12, 9, 3, 23, 1, 7, 7, 10, 13, 8, 0, 24, 5, 5, 2, 25, 4, 26, 4, 3

Offset: 1

Antti Karttunen as suggested by Don Reble, Oct 25 2022

nonn

a(n) gives the signature excitation of n (a concept proposed by Allan C. Wechsler, indicating the distance of n from the terms of A025487), when the primes in the "excited state", i.e., those present in A328478(n), are de-excited one by one, and the prime signature of n is preserved. See the example.

			For n = 98 = 2*7*7, the other 7 is de-excited as 7 -> 5 -> 3 -> 2, and the other 7 is de-excited as 7 -> 5 -> 3, to get 2*2*3 = 12 = A046523(98). There are 3+2 de-excitations in total, therefore a(98) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A025487 (positions of zeros), A046523, A056239.
Cf. also A319627, A328478, A358218.
Differs from A325799 for the first time at n=18, where a(18) = 1, while A325799(18) = 0.

{0}~Join~Array[Total@ Flatten[ConstantArray[PrimePi[#1], #2] & @@@ #] - Total@ Flatten@ MapIndexed[ConstantArray[First[#2], #1] &, ReverseSort[#[[All, -1]]]] &@ FactorInteger[#] &, 104, 2] (* Michael De Vlieger, Nov 02 2022 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A355930(n) = (A056239(n) - A056239(A046523(n)));

a(n) = A056239(n) - A356159(n) = A056239(n) - A056239(A046523(n)).

For all n, a(n) >= A358218(n). - Antti Karttunen, Nov 05 2022

A343172 Least number k such that the sum of the n Moebius function values beginning at k reaches the minimum value -A083544(n).

Original entry on oeis.org

2, 2, 29, 2, 101, 281, 429, 428, 2081, 6298, 30089, 30088, 143491, 567354, 693677, 693676, 8229, 693674, 1432677, 1123291, 1432677, 2156853, 25085909, 2156851, 25085909, 2156849, 24771577, 24771576, 126398226, 126398226, 3349160985, 389565283, 2928714078, 10441021690, 1353696733

Offset: 1

Don Reble, Apr 21 2021

nonn

Cf. A083544, A225420.

A341721 a(n) = minimum number of total votes needed for one party to win if there are n voters divided into equal districts.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 4, 6, 6, 6, 7, 8, 6, 8, 9, 8, 10, 9, 8, 12, 12, 10, 9, 14, 10, 12, 15, 12, 16, 14, 12, 18, 12, 14, 19, 20, 14, 15, 21, 16, 22, 18, 15, 24, 24, 18, 16, 18, 18, 21, 27, 20, 18, 20, 20, 30, 30, 21, 31, 32, 20, 24, 21, 24, 34, 27, 24, 24, 36, 25, 37, 38, 24

Offset: 1

Don Reble and N. J. A. Sloane, Feb 27 2021

nonn

This is a two-party election. The size d of each district must divide n, so there are d' = n/d equal districts. The districts are winner-takes-all, and tied districts go to neither candidate. For an even number of districts, it is enough to win half the districts and tie in one further district.
In general the best choice for d is not unique, since d and n/d give the same answer.
This is related to the gerrymandering question.
See A341578 for further information.

			For n=25 voters the smallest number of votes needed to win is 9: gerrymander 5 districts of 5 voters each, with three votes for the party in each of three districts.
For n=36 voters the smallest number of votes needed to win is 14: gerrymander 3 districts of 12 voters each, with seven votes for the party in each of two districts.
For n=64 voters the smallest number of votes needed to win is 24: gerrymander 8 districts of 8 voters each, with five votes for the party in each of four districts and four votes in a fifth district.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, How to use Gerrymandering to win with only 14 votes when your opponent has 22: (1) Locations of voters
N. J. A. Sloane, How to use Gerrymandering to win with only 14 votes when your opponent has 22: (2) Make three districts with 12 voters in each, drawn so that your 14 votes win 2 out of the 3 districts
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences: An illustrated guide with many unsolved problems, Guest Lecture given in Doron Zeilberger's Experimental Mathematics Math640 Class, Rutgers University, Spring Semester, Apr 28 2022: Slides; Slides (an alternative source).

See A341578 for the case when the number of voters must be a square.
See A341319 for a variant.
See also A290323.

with(NumberTheory);
f:=proc(n) local a,v,d,dp; a:=n;
for d in Divisors(n) do dp:=n/d;
if type(d,'even') and type(dp,'even')
  then v:=(d+2)*(dp)/4+d/2; if v

Mathematica

f[n_] := Module[{a, v, d, dp}, a = n; Do[dp = n/d; v = Which[ EvenQ[d] && EvenQ[dp], (d + 2)*(dp)/4 + d/2, EvenQ[d] && OddQ[dp], (d + 2)*(dp + 1)/4, OddQ[d] && EvenQ[dp], (d + 1)*(dp + 2)/4, OddQ[d] && OddQ[dp], (d + 1)*(dp + 1)/4]; If[v < a, a = v], {d, Divisors[n]}]; a]; Table[f[n], {n, 1, 100}] (* Jean-François Alcover, Apr 04 2023, after Maple code *)

from sympy import divisors
def A341721(n): return min((d+2-(d%2))*(e+2-(e%2))//4+int((d%2)or(e%2))-1 for d,e in ((v,n//v) for v in divisors(n) if v*v <= n)) # Chai Wah Wu, Mar 05 2021

a(n) is the minimum value over all ways of writing n = d*d' of:
  (d+1)*(d'+1)/4 if d and d' are both odd;
  (d+2)*(d'+1)/4 if d is even and d' is odd;
  (d+1)*(d'+2)/4 if d is odd and d' is even;
  (d+2)*(d'+2)/4-1 if d and d' are both even.
a(n) is bounded roughly between n/4 and n/2 (see graph). More precise bounds, which are attained infinitely often, are floor((n+1)/4 + sqrt(n)/2) <= a(n) <= floor((n/2)+1).

A323468 Primes p such that {1,2,3,4,5,6} coset-splits the cyclic multiplicative group modulo p, of order p-1.

Original entry on oeis.org

7, 13, 103, 487, 547, 769, 823, 967, 1063, 1249, 2521, 3049, 3109, 3187, 4327, 4447, 4483, 5167, 5881, 7027, 7207, 7477, 7933, 8221, 8293, 8461, 8527, 8803, 9181, 9721, 9769, 10837, 10867, 11083, 11503, 11527, 11743, 11887, 12043, 12049

Offset: 1

Don Reble, Jan 28 2019

nonn

S. K. Stein and S. Szabo, Algebra and Tiling, MAA Carus Monograph 25, 1994, page 77. (Beware omitted values.)

The Stein/Szabo sequence is A035030.

A291781 Consider the Watanabe tag system defined in A291067; a(n) = number of binary words of length n which terminate in a cycle.

Original entry on oeis.org

0, 0, 4, 12, 16, 32, 64, 128, 384, 704, 896, 2304, 4608, 7680, 16384, 36864, 65536, 143360, 303104, 565248, 1245184, 2473984, 4521984, 9961472, 19070976, 35389440, 78643200, 154664960, 289931264, 635437056, 1309671424, 2503999488, 5280628736

Offset: 1

Don Reble and N. J. A. Sloane, Sep 01 2017

nonn

Don Reble, Table of n, a(n) for n = 1..60

Cf. A284116, A291067, A291780.

A291780 Consider the Watanabe tag system defined in A291067; a(n) = number of binary words of length n which terminate at the empty word.

Original entry on oeis.org

2, 4, 4, 4, 16, 32, 64, 128, 128, 320, 1152, 1792, 3584, 8704, 16384, 28672, 65536, 118784, 221184, 483328, 851968, 1720320, 3866624, 6815744, 14483456, 31719424, 55574528, 113770496, 246939648, 438304768, 837812224, 1790967808, 3309305856

Offset: 1

Don Reble and N. J. A. Sloane, Sep 01 2017

nonn

Don Reble, Table of n, a(n) for n = 1..60

Cf. A284116, A291067, A291781.

A291779 a(n) = 2^n - 2^floor(2n/3).

Original entry on oeis.org

0, 1, 2, 4, 12, 24, 48, 112, 224, 448, 960, 1920, 3840, 7936, 15872, 31744, 64512, 129024, 258048, 520192, 1040384, 2080768, 4177920, 8355840, 16711680, 33488896, 66977792, 133955584, 268173312, 536346624, 1072693248, 2146435072, 4292870144, 8585740288, 17175674880, 34351349760, 68702699520

Offset: 0

Don Reble and N. J. A. Sloane, Sep 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..3318
Index entries for linear recurrences with constant coefficients, signature (2,0,4,-8).

Cf. A004523, A291778.

seq(2^n-2^floor(2*n/3),n=0..50); # Robert Israel, Sep 01 2017

LinearRecurrence[{2, 0, 4, -8}, {0, 1, 2, 4}, 37] (* Jean-François Alcover, Apr 02 2019 *)

G.f.: x/((1-2*x)*(1-4*x^3)). - Robert Israel, Sep 01 2017

A291778 a(n) = 2^floor(2*n/3).

Original entry on oeis.org

1, 1, 2, 4, 4, 8, 16, 16, 32, 64, 64, 128, 256, 256, 512, 1024, 1024, 2048, 4096, 4096, 8192, 16384, 16384, 32768, 65536, 65536, 131072, 262144, 262144, 524288, 1048576, 1048576, 2097152, 4194304, 4194304, 8388608, 16777216, 16777216, 33554432, 67108864, 67108864

Offset: 0

Don Reble and N. J. A. Sloane, Aug 31 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..4978
Index entries for linear recurrences with constant coefficients, signature (0,0,4).

Cf. A004523, A291779.

seq(2^floor(2*n/3),n=0..100); # Robert Israel, Sep 01 2017

LinearRecurrence[{0, 0, 4}, {1, 1, 2}, 41] (* Jean-François Alcover, Apr 02 2019 *)

a(n) = 2^A004523(n).
G.f.: (1+x+2*x^2)/(1-4*x^3). - Robert Israel, Sep 01 2017
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=0} 1/a(n) =  10/3.
Sum_{n>=0} (-1)^n/a(n) = 2/5. (End)

A236411 Let p(k) denote the k-th prime; a(n) = smallest p(m) > p(n) such that the n-2 differences between [p(n), p(n+1), ..., p(2n-2)] are the same as the n-2 differences between [p(m), p(m+1), ..., p(m+n-2)].

Original entry on oeis.org

5, 11, 13, 101, 37, 1277, 1279, 1616603, 57405419, 51448351, 76623356077, 115438255651991, 433241801791933

Offset: 2

Don Reble, Feb 05 2014

			n=5: We take the four primes [p(5)=11, 13, 17, 19], whose successive differences are 2, 4, 2. The next time we see this sequence of differences is at [101, 103, 107, 109], so a(5) = 101.

See A073615 for a very similar sequence.

(* This program generates the first ten terms of the sequence.  To generate more would require significantly greater computing resources *) dbp[n_]:=Differences[ Prime[ Range[ n,2n-2]]]; With[{prs=Prime[Range[ 3500000]]}, First/@ Flatten[ Table[Select[Partition[Drop[prs,n],n-1,1], Differences[#]==dbp[n]&,1],{n,2,11}],1]] (* Harvey P. Dale, Feb 05 2014 *)

A236411 = n->{d=vector(n-2,i,prime(n+i)-prime(n));
  forprime(p=prime(n+1),,
    for(k=1,#d, isprime(p+d[k])||next(2));
    for(k=1,#d, p+d[k]==nextprime(p+if(k>1,d[k-1])+1)||next(2));
  return(p))} \\ The second k-loop would suffice, but the first makes it 5x faster. Yields a(10), a(11) in ca. 3 sec (i7, 1.9Ghz). - M. F. Hasler, Feb 05 2014. [Erroneous ')' removed, Oct 09 2023]

Edited by N. J. A. Sloane, Feb 05 2014

cp's OEIS Frontend

User: Don Reble

A368374 a(n) = smallest k such that AM(k) - GM(k) >= n, where AM(k) and GM(k) are the arithmetic and geometric means of [1,...,k].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Extensions

A355930 Sum of the prime indices of n minus the sum of the prime indices of the smallest number with same prime signature as n, when the sum is taken with multiplicity, as in A056239.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343172 Least number k such that the sum of the n Moebius function values beginning at k reaches the minimum value -A083544(n).

Views

Author

Keywords

Crossrefs

A341721 a(n) = minimum number of total votes needed for one party to win if there are n voters divided into equal districts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A323468 Primes p such that {1,2,3,4,5,6} coset-splits the cyclic multiplicative group modulo p, of order p-1.

Views

Author

Keywords

References

Crossrefs

A291781 Consider the Watanabe tag system defined in A291067; a(n) = number of binary words of length n which terminate in a cycle.

Views

Author

Keywords

Links

Crossrefs

A291780 Consider the Watanabe tag system defined in A291067; a(n) = number of binary words of length n which terminate at the empty word.

Views

Author

Keywords

Links

Crossrefs

A291779 a(n) = 2^n - 2^floor(2n/3).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A291778 a(n) = 2^floor(2*n/3).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple